Problem 39 - College Physics

Question

A rocket fires two engines simultaneously. One produces a thrust of 725 \(\mathrm{N}\) directly forward, while the other gives a \(513-\mathrm{N}\) thrust at \(32.4^{\circ}\) above the forward, while the other gives a \(513-\mathrm{N}\) and direction (relative to the forward direction) of the resultant force that these engines exert on the rocket.

Step-by-Step Solution

Answer

The resultant force is approximately 1190 N at 13.42° above the forward direction.

Step 1 - Break forces into components

The thrust from each engine can be resolved into components. - The forward thrust of 725 N acts along the positive x-axis without any y-component.- For the second engine with a thrust of 513 N at 32.4°, compute x and y components: - x-component: \(F_{2x} = 513 \cos(32.4°)\) - y-component: \(F_{2y} = 513 \sin(32.4°)\)

Step 2 - Calculate components of the second force

Compute the components of the 513 N thrust using trigonometric functions:- \(F_{2x} = 513 \cos(32.4^{\circ}) \approx 433.66\text{ N}\)- \(F_{2y} = 513 \sin(32.4^{\circ}) \approx 275.27\text{ N}\)

Step 3 - Sum x and y components

Add up the x- and y-components to find the resultant force components:- Total x-component: \(F_{x} = 725 + 433.66 = 1158.66\text{ N}\)- Total y-component: \(F_{y} = 275.27\text{ N}\)

Step 4 - Calculate the magnitude of the resultant force

Use the Pythagorean theorem to find the magnitude of the resultant force:\[ F_{\text{resultant}} = \sqrt{(F_{x})^2 + (F_{y})^2} = \sqrt{(1158.66)^2 + (275.27)^2} \approx 1189.95\text{ N} \]

Step 5 - Find the direction of the resultant force

Compute the angle \(\theta\) with respect to the forward direction (x-axis) using the tangent function:\[ \theta = \arctan\left(\frac{F_{y}}{F_{x}}\right) = \arctan\left(\frac{275.27}{1158.66}\right) = 13.42^{\circ} \]

Key Concepts

Resultant Force

The concept of resultant force is essential in understanding how multiple forces acting on an object combine to produce a single effect.
When two engines of a rocket fire simultaneously, each engine generates its thrust, which can be thought of as a force with both magnitude and direction. The resultant force is a vector sum of these individual forces.
To find the resultant force, each force must be broken down into components along the axes of choice (usually, the horizontal and vertical axes).

• The horizontal component typically represents force parallel to the movement's direction.
• The vertical component represents the perpendicular force affecting the movement.

Once all the forces are resolved into their respective components, these can be summed separately for each axis.
By applying the Pythagorean theorem, the magnitude of the resultant force is calculated as:\[F_{\text{resultant}} = \sqrt{(F_{x})^2 + (F_{y})^2}\]This gives us a single vector that effectively represents all acting forces. Understanding resultant force helps predict the motion of the object under multiple influences.

Trigonometry in Physics

Trigonometry is a powerful tool used in physics for handling problems involving angles and directional forces.
In our rocket example, one engine thrusts at an angle of 32.4° to the forward direction. This angle plays a crucial role in determining how much force contributes to the forward motion and how much lifts the rocket vertically.
Trigonometric functions like cosine and sine are used to extract these components:

• Cosine function (\(\cos\)) is used to find the adjacent side (x-component) of the force:
\[ F_{2x} = 513 \cos(32.4^\circ) \]
• Sine function (\(\sin\)) is used to find the opposite side (y-component):
\[ F_{2y} = 513 \sin(32.4^\circ) \]

These functions allow us to decompose complex force vectors into manageable parts that align with our coordinate system.
By using trigonometry, we can better understand and compute the impact of forces applied at angles, ultimately allowing us to solve many real-world physics problems.

Vector Components

Vectors are quantities that have both magnitude and direction, crucial in physics for representing forces, velocities, and other directional quantities. Vector components simplify the process of vector addition.
In the exercise, each thrust from the rocket engines is described by a vector. To handle these vectors efficiently, they are broken down into components along chosen axes (commonly x and y).

• The x-component represents the horizontal influence of the vector.
• The y-component represents the vertical influence.

Vector components are calculated using trigonometric functions depending on the angle the vector makes with the axis. Once broken down, each component can be independently added or subtracted with corresponding components of other vectors.
The decomposition of vectors into components is particularly beneficial when multiple forces are acting at different angles. Summing these components provides an accurate representation of the net effect or resultant vector.
Understanding vector components is fundamental in predicting how forces will affect an object's movement, making it an indispensable technique in physics problem-solving.